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Résumé The problem Poincaré is concerned with is the achievement of an equilibrium between exactness and objectivity. The latter concerns a consensus with respect to empirical interpretation. His terminological tool to achieve the equilibrium process is the concept of “hypothesis“. Different forms of hypotheses characterize the hierarchy of sciences of which geometry holds the central position. The main thesis of this paper is that Poincaré's “apparent hypotheses“ (=conventions) have the same methodological function as Hilbert's axiom-schemata. Their reliability are not justified by a consistency proof but by a psycho-physiological reconstruction of a partially presupposed mathematical structure. His approach is an improvement of an attenuated version of ante rem structuralism that allows us to incorporate Poincaré's inheritance in the actual discussion in philosophy of mathematics. Note(s) Biographique(s) Gerhard Heinzmann est Prof
